arXiv research

Classifies ancient ovals in higher dimensional mean curvature flow.

The paper confirms conjectures about ancient ovals and provides counterexamples.

Enhanced estimates for ancient ovals and translators in 3D and 4D.

The paper classifies ovals in 4D space for a specific flow.

The paper classifies ancient ovals in higher dimensions and proves their symmetry and uniqueness.

Ancient ovals are key blowup limits in 3D Ricci flow near singularities.

Classifies ancient noncollapsed flows in 4D space.

Two ancient solutions to Gauss curvature flow are identified for cylinders.

We show that the only convex ancient solutions to curve shortening flow are the stationary lines, shrinking circles, Grim Reapers and Angenent ovals, completing the classification initiated by Daskalopoulos, Hamilton and Sesum and X.-J. Wang

Unique asymptotics found for special geometric flows.

We consider an embedded convex ancient solution $Γ_t$ to the curve shortening flow in $\mathbb{R}^2$. We prove that there are only two possibilities: the family $Γ_t$ is either the family of contracting circles, which is a type I ancient solution, or the family of evolving Angenent ovals, which correspond to a type II …

Study ancient flows in 4D, classifying based on bubble-sheet eigenvalues.

In this short article, we prove the existence of ancient solutions of the mean curvature flow that for t -> 0 collapse to a round point, but for t -> -infinity become more and more oval: near the center they have asymptotic shrinkers modeled on round cylinders S^j x R^n-j and near the tips they have asymptotic translat…

New convex ancient solutions found for flows by high powers of curvature.

We construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994. As time $t \rightarrow 0^-$ the solutions collapse to a round point where $0$ is the singular time. But as $t\rightarrow-\infty$ the solutions become more and more oval. Near the center the appropriately-resc…

New 1-parameter family of ovals identified in 4d Ricci flow classification.

In this article, we prove the mean convex neighborhood conjecture for the mean curvature flow of surfaces in $\mathbb{R}^3$. Namely, if the flow has a spherical or cylindrical singularity at a space-time point $X=(x,t)$, then there exists a positive $\varepsilon=\varepsilon(X)>0$ such that the flow is mean convex in a …

The study quantizes ancient flows in cylinders, revealing their asymptotic behavior.

It is a fundamental open problem for the mean curvature flow, and in fact for many partial differential equations, whether or not all blowup limits are selfsimilar. In this short note, we prove that for the mean curvature flow of mean convex surfaces all limit flows are selfsimilar (static, shrinking or translating) if…

In this paper, we prove the mean-convex neighborhood conjecture for neck singularities of the mean curvature flow in $\mathbb{R}^{n+1}$ for all $n\geq 3$: we show that if a mean curvature flow $\{M_t\}$ in $\mathbb{R}^{n+1}$ has an $S^{n-1}\times \mathbb{R}$ singularity at $(x_0,t_0)$, then there exists an $\varepsilon…

Resolves conjecture on cylindrical mean curvature flows in all dimensions.

A surface S in R^3 has the central plane oval property (cpo) if (i) S meets at least one affine plane transversally along a strictly convex oval, and (ii) Every such transverse oval on S has central symmetry. We show that a complete, connected C^2 surface with cpo must be either a generalized cylinder, or quadric. Appl…

Discussing rigidity properties of conics, inspired by billiards in ellipses.

Proof of Graustein's theorem in different geometries.

The Hessian Topology is a subject having interesting relations with several areas, for instance, differential geometry, implicit differential equations, analysis and singularity theory. In this article we study the problem of realization of a real plane curve as the Hessian curve of a smooth function. The plane curves …

In this paper we introduce the Constant Width Measure Set, which measures the constant width property of an oval, i.e. the planar simple closed strictly convex curve. We study its geometrical properties. We find the exact relation between the length and the area of the region bounded by an oval $M$. Namely, the followi…

Ancient curve flows classified into specific types.

Study of red blood cells using elastic surface theory.

Ancient solutions to mean curvature flow have unique shapes.

Study on ancient Ricci flows with positive curvature, proving noncollapsedness.

Ancient pancakes solve mean curvature flow problem.

Ancient solutions to Ricci flow in higher dimensions are mostly cylinders or solitons.

Ancient solutions on a strip are constant if polynomial, and have finite-dimensional space for slower growth.

Let $n\ge 3$ and $m=\frac{n-2}{n+2}$. We construct $5$-parameters, $4$-parameters, $3$-parameters ancient solutions of the equation $v_t=(v^m)_{xx}+v-v^m$, $v>0$, in $\mathbb{R}\times (-\infty,T)$ for some $T\in\mathbb{R}$. This equation arises in the study of Yamabe flow. We obtain various properties of the ancient so…

Ancient Ricci flow found from Taub-Bolt metric.

Ancient curves span halfplanes via flow.

Classifies ancient flows in a disc with boundary.

Classifies ancient convex curves in convex domains.

Ancient solutions to Kähler Ricci flow classified completely.

Ancient symplectic solutions to mean curvature flow are flat.

Ancient solutions to biharmonic heat equation bounded by polynomial dimensions.

Ancient Ricci flows are identified without curvature sign condition.

The paper constructs ancient solutions to curvature flows in bounded and unbounded regions.

Study ancient solutions to free boundary mean curvature flow in convex manifolds.

In this note we study the distribution of real inflection points among the ovals of a real non-singular hyperbolic curve of even degree. Using Hilbert's method we show that for any integers $d$ and $r$ such that $4\leq r \leq 2d^2-2d$, there is a non-singular hyperbolic curve of degree $2d$ in $\mathbb R^2$ with exactl…

New ancient solutions found for curvature flow in 2D.

Ancient flows by curvature powers in 2D have finite entropy.

Ancient solutions of Ricci flow with Type I growth are classified.